Which of the following is a solution to the equation \(2(\mathrm{x} + 1)^2 = \mathrm{x}^2 + 6\mathrm{x} + 5\)?

1. TRANSLATE the problem information

• Given equation: \(2(\mathrm{x} + 1)^2 = \mathrm{x}^2 + 6\mathrm{x} + 5\)
• Find: Which answer choice satisfies this equation

2. INFER the solution strategy

• The left side has a squared binomial that needs expanding
• After expansion, we'll have a quadratic equation to solve
• Strategy: Expand → Combine terms → Factor → Solve

3. SIMPLIFY by expanding the left side

• \(2(\mathrm{x} + 1)^2 = 2(\mathrm{x}^2 + 2\mathrm{x} + 1) = 2\mathrm{x}^2 + 4\mathrm{x} + 2\)
• Our equation becomes: \(2\mathrm{x}^2 + 4\mathrm{x} + 2 = \mathrm{x}^2 + 6\mathrm{x} + 5\)

4. SIMPLIFY by moving all terms to one side

• \(2\mathrm{x}^2 + 4\mathrm{x} + 2 - \mathrm{x}^2 - 6\mathrm{x} - 5 = 0\)
• Combine like terms: \(\mathrm{x}^2 - 2\mathrm{x} - 3 = 0\)

5. SIMPLIFY by factoring the quadratic

• Need two numbers that multiply to -3 and add to -2
• Those numbers are -3 and +1: \((-3)(+1) = -3\) and \((-3) + (+1) = -2\)
• Factor: \((\mathrm{x} - 3)(\mathrm{x} + 1) = 0\)

6. INFER the solutions using zero product property

• If \((\mathrm{x} - 3)(\mathrm{x} + 1) = 0\), then \(\mathrm{x} - 3 = 0\) or \(\mathrm{x} + 1 = 0\)
• Solutions: \(\mathrm{x} = 3\) or \(\mathrm{x} = -1\)
• Since \(\mathrm{x} = 3\) appears in the answer choices: \(\mathrm{x} = 3\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make expansion errors when computing \((\mathrm{x} + 1)^2\)

Many students incorrectly expand \((\mathrm{x} + 1)^2\) as \(\mathrm{x}^2 + 1\) (forgetting the middle term), leading to:

\(2(\mathrm{x}^2 + 1) = 2\mathrm{x}^2 + 2\) instead of \(2\mathrm{x}^2 + 4\mathrm{x} + 2\)

This creates the wrong equation \(2\mathrm{x}^2 + 2 = \mathrm{x}^2 + 6\mathrm{x} + 5\), which simplifies to \(\mathrm{x}^2 - 6\mathrm{x} - 3 = 0\). This quadratic doesn't factor nicely and produces solutions that don't match any answer choice, causing confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY reasoning: Sign errors when moving terms to one side of the equation

Students might write: \(2\mathrm{x}^2 + 4\mathrm{x} + 2 = \mathrm{x}^2 + 6\mathrm{x} + 5\) → \(2\mathrm{x}^2 + 4\mathrm{x} + 2 + \mathrm{x}^2 + 6\mathrm{x} + 5 = 0\)

This leads to \(3\mathrm{x}^2 + 10\mathrm{x} + 7 = 0\), which factors to approximately \(\mathrm{x} = -1.17\) or \(\mathrm{x} = -2\), neither matching the given choices.

The Bottom Line:

This problem tests careful algebraic manipulation through multiple steps. Success requires systematic expansion of binomial squares and meticulous attention to signs when combining terms.